First-Order Condition¶

Theorem¶

Let $a< b$ be real numbers, and let $f \colon (a, b) \to \mathbb{R}$ be a function. If $f$ is differentiable at $x=x_0\in (a, b)\text{,}$ and $f$ attains a local maximum or minimum at $x=x_0\text{,}$ then $f'(x_0) = 0\text{.}$

Interpretation¶

The first-order condition is a necessary criterion for local extrema: at an interior point where $f$ is differentiable, a local maximum or minimum forces a horizontal tangent, $f'(x_0)=0$ $0$ . It turns the search for extrema into solving the equation $f'(x)=0$ $0$ .

The reason is a sign argument. At a local maximum, $f(x)-f(x_0)\le0$ for all $x$ near $x_0$ , so the difference quotient $\frac{f(x)-f(x_0)}{x-x_0}$ is $\ge0$ for $x<x_0$ and $\le0$ for $x>x_0$ $x_{0}$ . Its left-hand limit is therefore $\ge0$ and its right-hand limit $\le0$ $0$ ; both equal $f'(x_0)$ , so $f'(x_0)=0$ $0$ . The case of a local minimum is the same with the signs reversed.

Each hypothesis is used. The point must be interior, so that $x$ can approach $x_0$ from both sides: $f(x)=x$ on $[0,1]$ has its minimum at $0$ and its maximum at $1$ , and its one-sided derivative is $1$ at both. And $f$ must be differentiable at $x_0$ ; otherwise an extremum can sit at a corner. This is why the candidates for extrema are the critical points together with the boundary points of the domain.

The condition is necessary, not sufficient: it only produces the stationary points as candidates, and each candidate still has to be checked by comparing $f(x)$ with $f(x_0)$ nearby. Later tools such as the second-order condition make this check easier, and Math IV uses the same condition for functions of several variables in the unconstrained extrema conditions.

Key Results¶

Local extrema of differentiable functions at interior points of the domain occur only at stationary points.
The condition is necessary but not sufficient for optimality.
Boundary points require separate analysis.

Examples¶

The function $f(x)=x^2$ satisfies $f'(0)=0$ at its minimum.
The function $f(x)=x^3$ satisfies $f'(0)=0$ but has no extremum there.

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 6

Aliases: FOC, first order condition, first-order necessary condition, Fermat's theorem on stationary points Keywords: interior extremum theorem Symbols: element of, real numbers

Last updated: 20 October 2026 03:02